In this paper we introduced a novel framework that we call MAgNet for solving PDEs on any mesh, possibly irregular. We proposed two variants of the architecture which gave promising results on benchmark datasets. We were effectively able to beat graph-based and grid-based architectures even when using the CNN variant of the proposed framework, therefore suggesting a novel way of adapting existing CNN architectures to make predictions on any mesh. The main added value of the proposed method is its very good performance on irregular meshes, including for super-resolution tasks, as it can be observed for the presented 1D and 2D experiments, when compared to SOTA methods. Notably, it seems to perform best when handed the smallest amount of data to work with (with a 64x64 training resolution), even in the condensed non-uniform case. This is a very desirable property for real-life data.
A limitation of our work however, is the significance of the learned interpolator. Indeed, compared with a simple cubic interpolation, the approach introduced here doesn't seem to offer a significant advantage and we leave improvement regarding this point for future work. Another improvement could be seen in the forecasting module. For now, MAgNet forecasts using a first-order explicit time-stepping scheme that is known to suffer from instability problems in numerical PDE and ODE solvers. Learned solvers seem to somehow circumvent this limitation even when using large time steps [1]}, [2]}, [3]}. In a future work, we wish to explore other time-stepping schemes such as the 4th order Runge-Kutta method [4]}, [5]} which is commonly used for solving PDEs. Finally, we observed that our MAgNet model appears to perform significantly better in a 1D than in a 2D setting (as shown in section REF ). While it is retaining its superiority in an irregular mesh setting, it appears a little less performant than the MPNN method, even with its GNN variant. Further research on why the regular setting poses issues is thus also reserved for future work.
